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Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations
Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation
1. | Laboratory Mathématiques et Informatique pour la Complexité et les Systèmes, CentraleSupélec, Univérsité Paris-Saclay, Campus de Gif-sur-Yvette, Plateau de Moulon, 3 rue Joliot Curie, 91190 Gif-sur-Yvette, France |
2. | Institute of Biological, Environmental and Rural Sciences, Aberystwyth University, Penglais Campus, Aberystwyth, Ceredigion, Wales, SY23 3DA, United Kingdom |
We relate together different models of non linear acoustic in thermo-elastic media as the Kuznetsov equation, the Westervelt equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) and estimate the time during which the solutions of these models keep closed in the $ L^2 $ norm. The KZK and NPE equations are considered as paraxial approximations of the Kuznetsov equation. The Westervelt equation is obtained as a nonlinear approximation of the Kuznetsov equation. Aiming to compare the solutions of the exact and approximated systems in found approximation domains the well-posedness results (for the Kuznetsov equation in a half-space with periodic in time initial and boundary data) are obtained.
References:
[1] |
S. I. Aanonsen, T. Barkve, J. N. Tjøtta and S. Tjøtta,
Distortion and harmonic generation in the nearfield of a finite amplitude sound beam, The Journal of the Acoustical Society of America, 75 (1984), 749-768.
|
[2] |
R. A. Adams, Sobolev Spaces, Vol. 65, Pure and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. |
[3] |
S. Alinhac,
Temps de vie des solutions régulières des équations d'Euler compressibles axisymétriques en dimension deux, Invent. Math., 111 (1993), 627-670.
doi: 10.1007/BF01231301. |
[4] |
S. Alinhac, A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations, Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002), (2002), Exp. No. Ⅰ, 33 pp. |
[5] |
N. S. Bakhvalov, Y. M. Zhileǐkin and E. A. Zabolotskaya, Nonlinear Theory of Sound Beams, American Institute of Physics Translation Series, American Institute of Physics, New York, 1987. |
[6] |
L. Bers, F. John and M. Schechter, Partial Differential Equations, Lectures in Applied Mathematics, 3A, American Mathematical Society, Providence, RI, 1979. |
[7] |
P. Caine and M. West,
A tutorial on the non-linear progressive wave equation (NPE). Part 2. Derivation of the three-dimensional cartesian version without use of perturbation expansions, Applied Acoustics, 45 (1995), 155-165.
|
[8] |
Z. Cao, H. Yin, L. Zhang and L. Zhu,
Large time asymptotic behavior of the compressible Navier-Stokes equations in partial space-periodic domains, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 1167-1191.
doi: 10.1016/S0252-9602(16)30061-3. |
[9] |
A. Celik and M. Kyed,
Nonlinear wave equation with damping: Periodic forcing and non-resonant solutions to the Kuznetsov equation, ZAMM Z. Angew. Math. Mech., 98 (2018), 412-430.
doi: 10.1002/zamm.201600280. |
[10] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Vol. 325, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 4$^th$ edition, Springer-Verlag, Berlin, 2016.
doi: 10.1007/978-3-662-49451-6. |
[11] |
A. Dekkers and A. Rozanova-Pierrat,
Cauchy problem for the Kuznetsov equation, Discrete Contin. Dyn. Syst., 39 (2019), 277-307.
doi: 10.3934/dcds.2019012. |
[12] |
A. Dekkers and A. Rozanova-Pierrat, Models of nonlinear acoustics viewed as an approximation of the Navier-Stokes and Euler compressible isentropicsystems, preprint, arXiv(1811.10850). |
[13] |
R. Denk, M. Hieber and J. Prüss,
Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.
doi: 10.1007/s00209-007-0120-9. |
[14] |
G. Di Blasio,
Linear parabolic evolution equations in $L^p$-spaces, Ann. Mat. Pura Appl., 138 (1984), 55-104.
doi: 10.1007/BF01762539. |
[15] |
M. Ghisi, M. Gobbino and A. Haraux,
Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation, Trans. Amer. Math. Soc., 368 (2016), 2039-2079.
doi: 10.1090/tran/6520. |
[16] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 24, Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. |
[17] |
B. Gustafsson and A. Sundström,
Incompletely parabolic problems in fluid dynamics, SIAM J. Appl. Math., 35 (1978), 343-357.
doi: 10.1137/0135030. |
[18] |
M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Academic Press, 1998.
![]() |
[19] |
D. Hoff,
Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14.
doi: 10.1007/BF00390346. |
[20] |
D. Hoff,
Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids, Arch. Rational Mech. Anal., 139 (1997), 303-354.
doi: 10.1007/s002050050055. |
[21] |
K. Ito,
Smooth global solutions of the two-dimensional Burgers equation, Canad. Appl. Math. Quart., 2 (1994), 283-323.
|
[22] |
F. John, Nonlinear Wave Equations, Formation of Singularities, Vol. 2, University Lecture Series, American Mathematical Society, Providence, RI, 1990.
doi: 10.1090/ulect/002. |
[23] |
P. M. Jordan,
An analytical study of Kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation, Phys. Lett. A, 326 (2004), 77-84.
doi: 10.1016/j.physleta.2004.03.067. |
[24] |
P. M. Jordan,
Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205.
doi: 10.3934/dcdsb.2014.19.2189. |
[25] |
B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst., (2011), 763–773. |
[26] |
B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 34 pp.
doi: 10.1142/S0218202512500352. |
[27] |
B. Kaltenbacher and V. Nikolić,
The Jordan-Moore-Gibson-Thompson equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time, Math. Models Methods Appl. Sci., 29 (2019), 2523-2556.
doi: 10.1142/S0218202519500532. |
[28] |
B. Kaltenbacher and M. Thalhammer,
Fundamental models in nonlinear acoustics part Ⅰ. Analytical comparison, Math. Models Methods Appl. Sci., 28 (2018), 2403-2455.
doi: 10.1142/S0218202518500525. |
[29] |
B. Kaltenbacher and I. Lasiecka,
An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; Exponential decay, Math. Nachr., 285 (2012), 295-321.
doi: 10.1002/mana.201000007. |
[30] |
T. Kato,
The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205.
doi: 10.1007/BF00280740. |
[31] |
V. P. Kuznetsov,
Equations of nonlinear acoustics, Soviet Phys. Acoust., 16 (1971), 467-470.
|
[32] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968. |
[33] |
M. J. Lesser and R. Seebass,
The structure of a weak shock wave undergoing reflexion from a wall, Journal of Fluid Mechanics, 31 (1968), 501-528.
|
[34] |
T. Luo, C. Xie and Z. Xin,
Non-uniqueness of admissible weak solutions to compressible Euler systems with source terms, Adv. Math., 291 (2016), 542-583.
doi: 10.1016/j.aim.2015.12.027. |
[35] |
S. Makarov and M. Ochmann,
Nonlinear and thermoviscous phenomena in acoustics, part Ⅱ, Acta Acustica United with Acustica, 83 (1997), 197-222.
|
[36] |
A. Matsumura and T. Nishida,
Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.
doi: 10.1007/BF01214738. |
[37] |
A. Matsumura and T. Nishida,
The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.
doi: 10.1215/kjm/1250522322. |
[38] |
B. E. McDonald, P. Caine and M. West,
A tutorial on the nonlinear progressive wave equation (NPE)–part 1, Applied Acoustics, 43 (1994), 159-167.
|
[39] |
B. E. McDonald and W. A. Kuperman,
Time-domain solution of the parabolic equation including nonlinearity, Comput. Math. Appl., 11 (1985), 843-851.
doi: 10.1016/0898-1221(85)90179-8. |
[40] |
S. Meyer and M. Wilke,
Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces, Evol. Equ. Control Theory, 2 (2013), 365-378.
doi: 10.3934/eect.2013.2.365. |
[41] |
A. Rozanova-Pierrat,
Qualitative analysis of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, Math. Models Methods Appl. Sci., 18 (2008), 781-812.
doi: 10.1142/S0218202508002863. |
[42] |
A. Rozanova-Pierrat,
On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media, Commun. Math. Sci., 7 (2009), 679-718.
doi: 10.4310/CMS.2009.v7.n3.a9. |
[43] |
A. Rozanova-Pierrat, Approximation of a compressible Navier-Stokes system by non-linear acoustical models, Proceedings of the International Conference DAYS on DIFFRACTION, (2015), 270–276. |
[44] |
T. C. Sideris,
Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys., 101 (1985), 475-485.
doi: 10.1007/BF01210741. |
[45] |
T. C. Sideris,
The lifespan of smooth solutions to the three-dimensional compressible Euler equations and the incompressible limit, Indiana Univ. Math. J., 40 (1991), 535-550.
doi: 10.1512/iumj.1991.40.40025. |
[46] |
T. C. Sideris, The lifespan of 3D compressible flow, Séminaire sur les Équations aux Dérivées Partielles, 5 (1992), 10 pp. |
[47] |
T. C. Sideris,
Delayed singularity formation in 2D compressible flow, Amer. J. Math., 119 (1997), 371-422.
doi: 10.1353/ajm.1997.0014. |
[48] |
M. F. Sukhinin,
On the solvability of the nonlinear stationary transport equation, Teoret. Mat. Fiz., 103 (1995), 23-31.
doi: 10.1007/BF02069780. |
[49] |
J. N. Tjøtta and S. Tjøtta, Nonlinear equations of acoustics, with application to parametric acoustic arrays, The Journal of the Acoustical Society of America, 69 (1981), 1644–1652. |
[50] |
P. J. Westervelt,
Parametric acoustic array, The Journal of the Acoustical Society of America, 35 (1963), 535-537.
|
[51] |
H. Yin and Q. Qiu,
The lifespan for 3-D spherically symmetric compressible sEuler equations, Acta Math. Sinica (N.S.), 14 (1998), 527-534.
doi: 10.1007/BF02580410. |
show all references
References:
[1] |
S. I. Aanonsen, T. Barkve, J. N. Tjøtta and S. Tjøtta,
Distortion and harmonic generation in the nearfield of a finite amplitude sound beam, The Journal of the Acoustical Society of America, 75 (1984), 749-768.
|
[2] |
R. A. Adams, Sobolev Spaces, Vol. 65, Pure and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. |
[3] |
S. Alinhac,
Temps de vie des solutions régulières des équations d'Euler compressibles axisymétriques en dimension deux, Invent. Math., 111 (1993), 627-670.
doi: 10.1007/BF01231301. |
[4] |
S. Alinhac, A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations, Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002), (2002), Exp. No. Ⅰ, 33 pp. |
[5] |
N. S. Bakhvalov, Y. M. Zhileǐkin and E. A. Zabolotskaya, Nonlinear Theory of Sound Beams, American Institute of Physics Translation Series, American Institute of Physics, New York, 1987. |
[6] |
L. Bers, F. John and M. Schechter, Partial Differential Equations, Lectures in Applied Mathematics, 3A, American Mathematical Society, Providence, RI, 1979. |
[7] |
P. Caine and M. West,
A tutorial on the non-linear progressive wave equation (NPE). Part 2. Derivation of the three-dimensional cartesian version without use of perturbation expansions, Applied Acoustics, 45 (1995), 155-165.
|
[8] |
Z. Cao, H. Yin, L. Zhang and L. Zhu,
Large time asymptotic behavior of the compressible Navier-Stokes equations in partial space-periodic domains, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 1167-1191.
doi: 10.1016/S0252-9602(16)30061-3. |
[9] |
A. Celik and M. Kyed,
Nonlinear wave equation with damping: Periodic forcing and non-resonant solutions to the Kuznetsov equation, ZAMM Z. Angew. Math. Mech., 98 (2018), 412-430.
doi: 10.1002/zamm.201600280. |
[10] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Vol. 325, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 4$^th$ edition, Springer-Verlag, Berlin, 2016.
doi: 10.1007/978-3-662-49451-6. |
[11] |
A. Dekkers and A. Rozanova-Pierrat,
Cauchy problem for the Kuznetsov equation, Discrete Contin. Dyn. Syst., 39 (2019), 277-307.
doi: 10.3934/dcds.2019012. |
[12] |
A. Dekkers and A. Rozanova-Pierrat, Models of nonlinear acoustics viewed as an approximation of the Navier-Stokes and Euler compressible isentropicsystems, preprint, arXiv(1811.10850). |
[13] |
R. Denk, M. Hieber and J. Prüss,
Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.
doi: 10.1007/s00209-007-0120-9. |
[14] |
G. Di Blasio,
Linear parabolic evolution equations in $L^p$-spaces, Ann. Mat. Pura Appl., 138 (1984), 55-104.
doi: 10.1007/BF01762539. |
[15] |
M. Ghisi, M. Gobbino and A. Haraux,
Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation, Trans. Amer. Math. Soc., 368 (2016), 2039-2079.
doi: 10.1090/tran/6520. |
[16] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 24, Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. |
[17] |
B. Gustafsson and A. Sundström,
Incompletely parabolic problems in fluid dynamics, SIAM J. Appl. Math., 35 (1978), 343-357.
doi: 10.1137/0135030. |
[18] |
M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Academic Press, 1998.
![]() |
[19] |
D. Hoff,
Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14.
doi: 10.1007/BF00390346. |
[20] |
D. Hoff,
Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids, Arch. Rational Mech. Anal., 139 (1997), 303-354.
doi: 10.1007/s002050050055. |
[21] |
K. Ito,
Smooth global solutions of the two-dimensional Burgers equation, Canad. Appl. Math. Quart., 2 (1994), 283-323.
|
[22] |
F. John, Nonlinear Wave Equations, Formation of Singularities, Vol. 2, University Lecture Series, American Mathematical Society, Providence, RI, 1990.
doi: 10.1090/ulect/002. |
[23] |
P. M. Jordan,
An analytical study of Kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation, Phys. Lett. A, 326 (2004), 77-84.
doi: 10.1016/j.physleta.2004.03.067. |
[24] |
P. M. Jordan,
Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205.
doi: 10.3934/dcdsb.2014.19.2189. |
[25] |
B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst., (2011), 763–773. |
[26] |
B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 34 pp.
doi: 10.1142/S0218202512500352. |
[27] |
B. Kaltenbacher and V. Nikolić,
The Jordan-Moore-Gibson-Thompson equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time, Math. Models Methods Appl. Sci., 29 (2019), 2523-2556.
doi: 10.1142/S0218202519500532. |
[28] |
B. Kaltenbacher and M. Thalhammer,
Fundamental models in nonlinear acoustics part Ⅰ. Analytical comparison, Math. Models Methods Appl. Sci., 28 (2018), 2403-2455.
doi: 10.1142/S0218202518500525. |
[29] |
B. Kaltenbacher and I. Lasiecka,
An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; Exponential decay, Math. Nachr., 285 (2012), 295-321.
doi: 10.1002/mana.201000007. |
[30] |
T. Kato,
The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205.
doi: 10.1007/BF00280740. |
[31] |
V. P. Kuznetsov,
Equations of nonlinear acoustics, Soviet Phys. Acoust., 16 (1971), 467-470.
|
[32] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968. |
[33] |
M. J. Lesser and R. Seebass,
The structure of a weak shock wave undergoing reflexion from a wall, Journal of Fluid Mechanics, 31 (1968), 501-528.
|
[34] |
T. Luo, C. Xie and Z. Xin,
Non-uniqueness of admissible weak solutions to compressible Euler systems with source terms, Adv. Math., 291 (2016), 542-583.
doi: 10.1016/j.aim.2015.12.027. |
[35] |
S. Makarov and M. Ochmann,
Nonlinear and thermoviscous phenomena in acoustics, part Ⅱ, Acta Acustica United with Acustica, 83 (1997), 197-222.
|
[36] |
A. Matsumura and T. Nishida,
Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.
doi: 10.1007/BF01214738. |
[37] |
A. Matsumura and T. Nishida,
The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.
doi: 10.1215/kjm/1250522322. |
[38] |
B. E. McDonald, P. Caine and M. West,
A tutorial on the nonlinear progressive wave equation (NPE)–part 1, Applied Acoustics, 43 (1994), 159-167.
|
[39] |
B. E. McDonald and W. A. Kuperman,
Time-domain solution of the parabolic equation including nonlinearity, Comput. Math. Appl., 11 (1985), 843-851.
doi: 10.1016/0898-1221(85)90179-8. |
[40] |
S. Meyer and M. Wilke,
Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces, Evol. Equ. Control Theory, 2 (2013), 365-378.
doi: 10.3934/eect.2013.2.365. |
[41] |
A. Rozanova-Pierrat,
Qualitative analysis of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, Math. Models Methods Appl. Sci., 18 (2008), 781-812.
doi: 10.1142/S0218202508002863. |
[42] |
A. Rozanova-Pierrat,
On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media, Commun. Math. Sci., 7 (2009), 679-718.
doi: 10.4310/CMS.2009.v7.n3.a9. |
[43] |
A. Rozanova-Pierrat, Approximation of a compressible Navier-Stokes system by non-linear acoustical models, Proceedings of the International Conference DAYS on DIFFRACTION, (2015), 270–276. |
[44] |
T. C. Sideris,
Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys., 101 (1985), 475-485.
doi: 10.1007/BF01210741. |
[45] |
T. C. Sideris,
The lifespan of smooth solutions to the three-dimensional compressible Euler equations and the incompressible limit, Indiana Univ. Math. J., 40 (1991), 535-550.
doi: 10.1512/iumj.1991.40.40025. |
[46] |
T. C. Sideris, The lifespan of 3D compressible flow, Séminaire sur les Équations aux Dérivées Partielles, 5 (1992), 10 pp. |
[47] |
T. C. Sideris,
Delayed singularity formation in 2D compressible flow, Amer. J. Math., 119 (1997), 371-422.
doi: 10.1353/ajm.1997.0014. |
[48] |
M. F. Sukhinin,
On the solvability of the nonlinear stationary transport equation, Teoret. Mat. Fiz., 103 (1995), 23-31.
doi: 10.1007/BF02069780. |
[49] |
J. N. Tjøtta and S. Tjøtta, Nonlinear equations of acoustics, with application to parametric acoustic arrays, The Journal of the Acoustical Society of America, 69 (1981), 1644–1652. |
[50] |
P. J. Westervelt,
Parametric acoustic array, The Journal of the Acoustical Society of America, 35 (1963), 535-537.
|
[51] |
H. Yin and Q. Qiu,
The lifespan for 3-D spherically symmetric compressible sEuler equations, Acta Math. Sinica (N.S.), 14 (1998), 527-534.
doi: 10.1007/BF02580410. |
KZK | NPE | Westervelt | ||||
periodic boundary condition problem | initial boundary value problem | viscous and inviscid case | viscous case | inviscid case | ||
Theorem | Theorem 2.1 | Theorem 2.2 | Theorem 3.1 | Theorem 4.2 | ||
Derivation | paraxial approximation |
paraxial approximation |
||||
Approximation domain | the half space |
|||||
Approximation order | ||||||
Estimation | ||||||
Initial data regularity | for |
for |
for |
for |
for |
|
Data regularity for remainder boundness | for |
for for |
for for |
for |
for |
KZK | NPE | Westervelt | ||||
periodic boundary condition problem | initial boundary value problem | viscous and inviscid case | viscous case | inviscid case | ||
Theorem | Theorem 2.1 | Theorem 2.2 | Theorem 3.1 | Theorem 4.2 | ||
Derivation | paraxial approximation |
paraxial approximation |
||||
Approximation domain | the half space |
|||||
Approximation order | ||||||
Estimation | ||||||
Initial data regularity | for |
for |
for |
for |
for |
|
Data regularity for remainder boundness | for |
for for |
for for |
for |
for |
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